To calculate string $S$-matrix, we mainly use Faddeev-Popov gauge fixing method, as in chapter 6 of Polchinsky's book 《string theory》. But in section 6.2, 'tree level amplitude', I didn't find that it calculates the path integral corresponding to the ghost fields, it only calculates the generating functional corresponding to the matter fields, so i still don't know how to calculate the string path integral to compute tree level amplitude. Can any one elaborate that?

The following is from .section 6.2 in Polchinsky 《String theory》: enter image description here if you have the book in your hand you can look at it. I think that generating functional is only for ordinary field but does not deal with ghost field. in the next section 6.3 it starts dealing with calculating particular vertex operator expectation value using this generating functional and never compute ghost fields.

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    $\begingroup$ Can you be more explicit about what you want? The standard expression I know for the stringy S-matrix is a gauge-fixed version that does integrate over the ghost fields. Perhaps writing down the expressions (and an explanation of the notation used) would make it clearer what the issue here is. $\endgroup$
    – ACuriousMind
    Oct 28 '17 at 10:32
  • $\begingroup$ Could you elaborate on the last sentence (v3)? The next section 6.3 is precisely devoted to the $bc$ ghost system. $\endgroup$
    – Qmechanic
    Nov 5 '17 at 17:38

Your doubt is actually a symptom that you need to go back to the chapter 5. Ghosts insertions are related to moduli (b-ghosts) and fixing vertex operators (c-ghosts). In chapter 5, section 5.3, in the Riemann-Roch theorem proof, he shows that in order to conserve the ghost number the number of b-ghost insertions minus the number of c-ghost insertions should be $-3/2\chi$, where $\chi$ is the Euler number of the manifold. Then the equation 5.3.18 gives the value of the Faddev-Popov measure for those insertions, the ghost part of the path integral. In chapter 6, section 6.3, he work out all this explicitly for the $S_2$ for arbitrary numbers of bc-ghost obeying total ghost number $3/2\chi$.

The example given in section 6.3 $$ \langle c(z_1)c(z_2)c(z_3)\tilde{c}(z_4)\tilde{c}(z_5)\tilde{c}(z_6)\rangle_{S_2}=C_{S_2}^{g}\left\| \matrix{1&1&1 \\z_1&z_2&z_3\\z_1^{2}&z^{2}_{2}&z^{2}_{3}}\right\|\left\| \matrix{1&1&1 \\\tilde{z}_4&\tilde{z}_5&\tilde{z}_6\\\tilde{z}_4^{2}&\tilde{z}^{2}_{5}&\tilde{z}^{2}_{6}}\right\|\implies$$ $$\implies\langle c(z_1)c(z_2)c(z_3)\tilde{c}(z_4)\tilde{c}(z_5)\tilde{c}(z_6)\rangle_{S_2}=\pm C_{S_2}^{g}z_{12}z_{23}z_{13}\tilde{z}_{45}\tilde{z}_{46}\tilde{z}_{56} $$ where the determinants comes from the determinant $\det(\mathcal{C}_{0j}^{a}(\sigma_i))$ of the equation 5.3.18.

Latter in the chapter 6 we see that this ghost insertions are necessary in order to the amplitudes be independent of the positions of the unintegrated vertex operators. They are there to cancel the denominators $(z_{12}z_{23}z_{13}\tilde{z}_{45}\tilde{z}_{46}\tilde{z}_{56})^{-1}$ produced by the unintegrated vertex operators.


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